An important part of categorical universal algebra is played by monads, a point of view that we have not considered here.
Indeed, Lawvere theories can be seen as finitary monads [1]}, with the category of algebras giving the associated variety. This connection has been a fruitful one, relating areas of research that are, on the surface, very different, see e.g. [2]}, [3]}, [4]}. A natural question is whether there is an analogous approach for partial algebraic theories. We conjecture that there is, with certain formal monads [5]} in the 2-category \(\mathsf {DCRC}^\le \)  playing the role of finitary monads.
We expect that other constructions of categorical universal algebra (e.g. [6]}, [7]}) will have corresponding partial accounts.
